SOLUTION: Please help and check what I have so far: the question: The function p is a fourth-degree polynomial with x-intercepts 1, 4, and 10 and y-intercept -2. If p(x) is positive only

Question 916703: Please help and check what I have so far:
the question: The function p is a fourth-degree polynomial with x-intercepts 1, 4, and 10 and y-intercept -2. If p(x) is positive only on the interval (4, 10), find p(x).
p(x)= (x-1)(x-4)(x-10)
I don't understand where to put -2 or what to do with the information about the positive only in the interval (4,10)
Thank you
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
p is a fourth degree polynomial, but you only have 3 roots. So one root has to be repeated (you need 4 total roots).

The y-intercept is -2, so that means p(0) = -2

So you may have p(x)= k(x-1)^2(x-4)(x-10) or p(x)= k(x-1)(x-4)^2(x-10) or p(x)= k(x-1)(x-4)(x-10)^2

Let's see what happens with p(x)= k(x-1)^2(x-4)(x-10)

p(x)= k(x-1)^2(x-4)(x-10)

p(0)= k(0-1)^2(0-4)(0-10)

-2 = 40k

40k = -2

k = -2/40

k = -1/20

So the function is

This graph (in green) confirms it

Let me know if you need more help or if you need me to explain a step in more detail.
Feel free to email me at jim_thompson5910@hotmail.com
or you can visit my website here: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim
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